Astronomy Answers
Coincidence Periods

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1. Introduction

At least 2500 years ago astronomers in Babylon discovered that solar eclipses and lunar eclipses often recur under similar circumstances after 223 synodical months (counted according to the phases of the Moon, not according to the western calendar) or about 6585⅓ days. This period is nowadays called the saros. We call a series of eclipses that each occur one saros after the previous one a saros series.

To get a solar eclipse or lunar eclipse, two independents series of periodic phenomena must come together, namely the proper phase of the Moon (New Moon for a solar eclipse, or Full Moon for a lunar eclipse) and passage of the Moon through a node of the orbit of the Moon around the Earth.

2. Strictly Periodic Phenomena

We study the coming together of two different periodic phenomena A and B. When they coincide then we have the combination phenomenon which we'll call Z. We assume that phenomenon A has a period \( P_A \), phenomenon B a period \( P_B \), and that the ratio of those periods is a rational number:

\begin{equation} \frac{P_A}{P_B} = γ' = \frac{a}{b} \end{equation}

with \( a \) and \( b \) whole numbers. That means that \( b \) periods \( P_A \) are exactly as long as \( a \) periods \( P_B \). That long after a previous Z there will be a next Z. We call the period \( y = b P_A \) the prediction period. We assume that \( a \) and \( b \) have no factors in common. If they do, then you can just divide them out.

If an A and a B happen at the same time then we get the combination phenomenon Z. Often there is a little leeway so that A and B may have a small time difference and yet give a Z. We can find two boundaries \( d_1 \) and \( d_2 \) so that a Z certainly occurs if the time difference between an A and the closest B is less than \( d_1 P_B \), and a Z certainly does not occur if the time difference is more than \( d_2 P_B \).

If you go \( y \) forward or backward in time from a Z then you find another Z. All of those phenomena Z form a collection that we call a prediction series. We call the collection of phenomena A that is associated with such a prediction series a prediction series (of A) as well, and likewise for B. In a prediction series of A or B or Z, the different phenomena of the same kind are all \( y \) apart.

We'll ignore the prediction series of B below, but everything works the same way for Bs as for As, except that you have to use some special values that hold for Bs instead of the corresponding values that hold for As.

We can associate each A with a prediction series. There are then \( b \) different prediction series. The phenomena Z won't happen in all of those prediction series.

We first assign an ordinal number to each A that is 1 greater for each than the number of the previous A. We use \( k \) to denote the ordinal number of A. A suitable formula for calculating the number of the prediction series is then

\begin{equation} s ≡ a k + s_0 \pmod{b} \label{eq:sarosnummer} \end{equation}

where the \( \bmod b \) means "except for multiples of \( b \)", and the \( s_0 \) is the number of the series that \( k \) = 0 belongs to. If you want to go the other way and find which \( k \) belong to the series with number \( s \), then you can use

\begin{equation} k = l (s - s_0) + n b \label{eq:ks} \end{equation}

where \( l \) is a number such that \( a l ≡ 1 \pmod{b} \) and \( n \) is an ordinal number that counts phenomena A in a prediction series. In general,

\begin{equation} l ≡ a^{φ(b) - 1} \pmod{b} \end{equation}

with \( φ(b) \) the number of non-negative whole numbers less than \( b \) that are prime relative to \( b \).

The condition of phenomena B at the moment of an A is

\begin{equation} S'_k = S_0 + γ' k \label{eq:b-toestand} \end{equation}

If \( S'_k \) is a whole number, then we are exactly in the middle of a B, and thus also exactly in the middle of a Z. \( S_0 \) is the condition of B at the moment that corresponds to \( k \) = 0. With the definition

\begin{equation} s'_* = a k + s_0 - b S'_k \label{eq:bestenummer} \end{equation}

we find that

\begin{equation} s'_* = s_0 - b S_0 \end{equation}

which is a constant. The whole number closest to \( s'_* \) is the number (except for multiples of \( b \)) of the series for which A and B are closest together. We refer to that series as the best series.

If we insert equation \ref{eq:ks} into equation \ref{eq:b-toestand} then we find

\begin{equation} S'_k = S_0 + \frac{a}{b} l (s - s_0) + a n = S_0 - \frac{a}{b} s_0 + \frac{a l}{b} s + a n \end{equation}

If we modify \( n \) by adding \( ∆n \) to it, or \( s \) by adding \( ∆s \) to it, then \( S'_k \) gets \( ∆S'_k \) added to it. Only the difference of \( S'_k \) from the nearest whole number is important for determining whether a Z will happen, so we can calculate \( \bmod 1 \). Then we find

\begin{equation} ∆S'_k = \frac{a l}{b} ∆s + a ∆n = \frac{1}{b} ∆s \pmod{1} \label{eq:delta-s} \end{equation}

The term \( a ∆n \) disappears because it is always a whole number, and \( \dfrac{a l}{b} ≡ \dfrac{1}{b} \pmod{1} \) because \( a l ≡ 1 \pmod{b} \) by definition.

If the \( S'_k \) of a prediction series differs less than \( d_1 \) from a whole number, then there is certainly a Z in that series. If the difference lies between \( d_1 \) and \( d_2 \), then there is sometimes a Z, and if the difference is greater than \( d_2 \), then there is certainly no Z. The best prediction series has \( s = [s'_*] \). For that series, \( S'_k \) is closest to a whole number. It follows from equation \ref{eq:delta-s} that there is a Z for every A in prediction series for which \( s \) lies between \( s'_* - d_1 b \) and \( s'_* + d_1 b \) (inclusive). For prediction series between \( s'_* - d_2 b \) and \( s'_* - d_1 b - 1 \) or between \( s'_* + d_1 b + 1 \) and \( s'_* + d_2 b \) there is a Z for some but not all A. If \( s \) is outside of those ranges, then there is no Z in that series.

3. Almost Periodic Phenomena With Fixed Periods

Up till now we've assumed that the ratio of the periods of A and B was a rational number (a division involving only whole numbers) which we knew exactly. If one or more of these assumptions is not met, then we can still approximate the ratio with a rational number, but then our approximation won't quite fit the reality. We call the real ratio of the periods \( γ \) and continue to use \( γ' \) for our approximation of the ratio:

\begin{align} γ \| = \dfrac{P_A}{P_B} \\ γ' \| = \dfrac{a}{b} \end{align}

We call the difference between the approximation and the real ratio \( δ \):

\begin{equation} δ = γ' - γ = \dfrac{a}{b} - γ \end{equation}

We can continue to use equation \ref{eq:sarosnummer} to calculate the number of the prediction series. For the condition of B at the time of an A we must adjust equation \ref{eq:b-toestand} to

\begin{equation} S_k = S_0 + γ k \label{eq:b-toestand-2} \end{equation}

and we adjust equation \ref{eq:bestenummer} for the number of the best prediction series to

\begin{equation} s_* = a k + s_0 - b S_k \label{eq:bestenummer-2} \end{equation}

from which

\begin{equation} s_* = b δ k + s_0 - b S_0 = b δ k + s'_* \end{equation}

Now \( s_* \) is no longer a constant, so the range of prediction series in which phenomena Z occur shifts with time at a rate of 1 per period \( \dfrac{P_A}{b δ} \).

The prediction series that yield phenomena Z around a given date won't do that indefinitely. If they yield phenomena Z then we call them active and otherwise inactive. Each prediction series will switch between active and inactive and back again in a grand period which we'll indicate as \( c \) and which is equal to

\begin{equation} c = \dfrac{P_A}{δ} \end{equation}

With this, equation \ref{eq:delta-s} becomes

\begin{equation} ∆S_k = l \left( \dfrac{a}{b} - δ \right) ∆s + (a - b δ) ∆n = \left( \dfrac{1}{b} - l δ \right) ∆s - b δ ∆n \pmod{1} \label{eq:sk-bijna} \end{equation}

If \( b δ ∆n \) is less than \( d_1 \), then there is certainly a Z, and if it is greater than \( d_2 \) then there is certainly no Z. It follows that an active period of a prediction series contains between \( \dfrac{2 d_1}{b δ} \) and \( \dfrac{2 d_2}{b δ} \) phenomena Z.

If you go to a different \( s \) then you go not just to a different prediction series but also (with equation \ref{eq:ks}) to a different \( k \), hence to a different time. This was not important in the previous section, because there the prediction series did not depend on time (because the prediction period was exactly right), but here this is important. We want to see what happens to \( S_k \) if you go to a different prediction series without going to a different date. We can do this, because equation \ref{eq:ks} shows that \( ∆k = l ∆s + b ∆n \) so \( ∆k = 0 \) if \( ∆n = -\dfrac{l}{b} ∆s \). If we insert that into equation \eqref{eq:sk-bijna} then we find, like for strictly periodic phenomena,

\begin{equation} ∆S_k = \dfrac{1}{b} ∆s \pmod{1} \end{equation}

This means that between \( 2 d_1 b \) and \( 2 d_2 b \) prediction series are active at any one time.

Because prediction series now switch between active and inactive, it is interesting to give each active period its own prediction series number. We can do that, because equation \ref{eq:sarosnummer} has \( \bmod b \) in it, so we can choose to given each active period from a given prediction series a number that is \( b \) greater than that of the previous active period. For example, like this:

\begin{equation} s = \left( \left( a k + s_0 - \left\lfloor s_* - \dfrac{b}{2} \right\rfloor \right) \bmod b \right) + \left\lfloor s_* - \dfrac{b}{2} \right\rfloor \label{eq:sarosnummer-2} \end{equation}

Because \( (x×y) \bmod b = ((x \bmod b)×(y \bmod b)) \bmod b \), you may replace \( a k \) from equation \ref{eq:sarosnummer-2} with \( (a \bmod b)×(k \bmod b) \), which may be useful to keep the calculation numbers as small as possible when \( k \) becomes very large.

If \( s_* \) is equal to a whole number, then A and B exactly coincide, so then we've found the middle of the active period. \( s_* \) is equal to a particular \( s \) (a whole number) if \( k \) is equal to

\begin{equation} k_s = \dfrac{s - s_0 + b S_0}{b δ} = \dfrac{s - s'_*}{b δ} \end{equation}

This \( k_s \) need not be a whole number and need not even fit in prediction series \( s \), but the \( k \) from series \( s \) that is closest to \( k_s \) indicates the middle of that series. With this, we can assign ordinal numbers

\begin{equation} m = \left[ \dfrac{k - k_s}{b} \right] \label{eq:m} \end{equation}

to the different Z from a prediction series. In that formula, \( k \) must fit the prediction series \( s \) (e.g., using equation \ref{eq:ks}). If \( m \) is equal to 0, then we've found the middle of the active period. If \( m \) is greater than 0, then we are past the middle, and if \( m \) is less than 0, then we are still before the middle.

Equation \ref{eq:m} depends in two different ways on \( k \): once because it contains \( k \) explicitly, and once because \( k_s \) depends on \( k \) via \( s \). Using equation \ref{eq:sarosnummer} we can say a bit more about how \( m \) depends on \( k \).

\begin{align} m' \| ≡ \dfrac{k - k_s}{b} \\ m \| = [m'] \end{align}

\begin{equation} \begin{split} m' \| = \dfrac{k - k_s}{b} \\ \| = \dfrac{k - \dfrac{s - s'_*}{bδ}}{b} \\ \| = \dfrac{k - \dfrac{ak + s_0 \bmod b - s'_*}{bδ}}{b} \\ \| = \dfrac{k - \dfrac{ak + s_0 - s'_*}{bδ}}{b} \bmod \dfrac{1}{bδ} \\ \| = \left( 1 - \dfrac{a}{bδ} \right) \dfrac{k}{b} + \dfrac{s'_* - s_0}{b^2δ} \bmod \dfrac{1}{bδ} \\ \| = -\dfrac{γk}{bδ} + \dfrac{s'_* - s_0}{b^2δ} \bmod \dfrac{1}{bδ} \\ \| = \dfrac{-γk + \dfrac{s'_* - s_0}{b} \bmod 1}{bδ} \end{split} \label{eq:m'} \end{equation}

4. Almost Periodic Phenomena With Varying Periods

Up till now we've assumed that the ratio of periods is constant, but in practice the ratio usually slowly varies with time. If the variation is slow, then we can still use reasonably simple formulas. We assume that the ratio \( γ \) is not a constant but varies according to

\begin{equation} γ_k = γ_0 + γ_1 k \end{equation}

The only important adjustments relative to the case of constant periods are to formulas that feature \( δ \) or \( S_k \). We find

\begin{align} δ \| = \dfrac{a}{b} - γ_0 - γ_1 k = δ_0 - γ_1 k \\ S_k \| = S_0 + γ_0 k + \dfrac{1}{2} γ_1 k^2 \\ s_* \| = a k + s_0 - S_k = b δ_0 k - \dfrac{1}{2} b γ_1 k^2 + s'_* \end{align}

To find the best Z of prediction series \( s \) we must now solve a quadratic equation:

\begin{equation} -\dfrac{1}{2} b γ_1 k_s^2 + b δ_0 k_s + s'_* - s = 0 \end{equation}

The solution is

\begin{equation} k_s = \dfrac{δ_0}{γ_1} \left( 1 - \sqrt{1 - \dfrac{2γ_1}{bδ_0^2}(s - s'_*)} \right) \end{equation}

If \( |s - s'_*| \) is much less than \( \left| \dfrac{bδ_0^2}{2γ_1} \right|| \), then the solution is approximately equal to

\begin{align} k_s \| = \dfrac{s - s'_*}{bδ_0} + \dfrac{γ_1 (s - s'_*)^2}{2b^2δ_0^3} \\ k_s \| ≡ k_{s0} + k_{s1} \\ k_{s0} \| ≡ \dfrac{s - s'_*}{bδ_0} \\ k_{s1} \| ≡ \dfrac{γ_1 (s - s'_*)^2}{2b^2δ_0^3} \end{align}

The calculation of \( m' \) then goes in a couple of steps:

\begin{align} m' \| = m_0' + m_1' \\ m_0' \| = \dfrac{k - k_{s0}}{b} \notag \\ \| = -\dfrac{γ_0k}{bδ_0} + \dfrac{s'_* - s_0}{b^2δ_0} \bmod \dfrac{1}{bδ_0} \notag \\ \| = \dfrac{-γ_0k + \dfrac{s'_* - s_0}{b} \bmod 1}{bδ_0} \\ m_1' \| = -\dfrac{k_{s1}}{b} = -\dfrac{γ_1(k - bm_0')^2}{2bδ_0} \end{align}

So first calculate \( m' \) from equation \ref{eq:m'} as if \( γ = γ_0 \) and call it \( m_0' \), and then calculate \( m_1' \) as a correction to \( m_0' \).

5. Related Phenomena

Suppose you can find for each phenomenon A a related phenomenon A′ which happens \( ε P_A \) later than A, with \( ε a \) a whole number. Then you'll find the same kind of behavior for the combination phenomena Z′ of A′ and B as for the combination phenomena Z of A and B, and you can use the same formulas for A′ and Z′ as for A and Z, except that some constants are different. You could, for example, insert \( k + ε \) in place of plain \( k \) into the formulas for A and Z. The formula to calculate the number of the prediction series then becomes

\begin{equation} s′ = a (k + ε) + s_0 = a k + (s_0 + a ε) \bmod b \end{equation}

You then find that the best prediction series have the exact same number (\( s'_* \)) for strictly periodic Z and Z′, so (almost) the same range of prediction series numbers is active for Z and for Z′.

6. The Next Events

Suppose that event A or Z occurs for a particular \( k \) corresponding to a particular \( s \) and \( m \). Then when are the next interesting events A or A' or Z or Z'?

• The next A (for \( k + 1 \)) then belongs to saros \( s + a \bmod b \) and has approximately \( m + \dfrac{γb}{δ} \).
• The next A that belongs to the same saros \( s \) then has number \( k + b \) and has \( m + 1 \).
• The next A that belongs to the next saros \( s + 1 \) then has number \( k + 1 \) and has approximately \( m + \dfrac{γl}{bδ} \bmod \dfrac{1}{bδ} \).
• The next A that belongs to the previous saros \( s - 1 \) then has number \( k + b - l \) and has approximately \( m + \dfrac{γ×(b - l)}{bδ} \bmod \dfrac{1}{bδ} \), which is approximately \( ±1 \)
.
• The A' that is related to A (and thus occurs a half \( P_A \) later for the same \( k \)) belongs to A-saros \( s + aε \bmod b \).
• Suppose that a Z occurred at this A. When is the next Z? Look at the first couple of multiples of \( a \), modulus \( b \). If those are close to 0 (or to \( b \)) then there is a chance of another Z. If \( a \bmod b ≤ 2d_2b \), then the very next A might be a Z again (if the previous Z was sufficiently early in its series, i.e., if \( m \) is quite negative). After that, \( ∆k = \dfrac{b}{a \bmod b} \) is a good candidate.

These rules are summarized in Table 1.

Table 1: Next Events

7. Comparison with the Eclipse Calculations by Fred Espenak

The "Catalog of Solar Eclipses" of November 1999 by Fred Espenak lists all 11897 solar eclipses during the years from −1999 to +3000, including their dates and saros numbers. The "Catalog of Lunar Eclipses" of November 1999 by Fred Espenak lists all 12186 lunar eclipses in the same period, including their dates and saros numbers.

(For the solar eclipses I noticed that the saros numbers of 4 of them in the catalog are wrong: the saros numbers of the 512th, 649th, 3627th, and 7553rd solar eclipse from the catalog are equal to 0 but should have been −9, −4, 33, and 91. The solar eclipses are sorted by date in the catalog, but the lunar eclipse aren't ― at least not perfectly.)

I calculated \( k \) (from the date) and also \( m \) and \( s \) for all of those eclipses, using the formulas given above. "My" saros number \( s \) is equal to that from the catalogs, except for the four solar eclipses that I mentioned. For succesive solar eclipses and successive lunar eclipses from the catalogs I find only the following combinations of \( ∆s \), \( ∆k \), and \( ∆m \):

After every solar eclipse there is a lunar eclipse ½ or 5½ months later. For the first lunar eclipse after each solar eclipse I find (with each \( ∆x = x_\text{Moon} - x_\text{Sun} \))

Each lunar eclipse is followed by a solar eclipse ½ or 5½ months later. For the first solar eclipse after each lunar eclipse I find (with each \( ∆x = x_\text{Moon} - x_\text{Sun} \))

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Last updated: 2021-07-19